Advanced quantum mechanics; materials and photons

Graduate Texts in Physics

For further volumes:
/>
www.pdfgrip.com


Graduate Texts in Physics
Graduate Texts in Physics publishes core learning/teaching material for graduate- and
advanced-level undergraduate courses on topics of current and emerging fields within
physics, both pure and applied. These textbooks serve students at the MS- or PhD-level
and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently
to serve as required reading. Their didactic style, comprehensiveness and coverage of
fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.

Series Editors
Professor Richard Needs
Cavendish Laboratory
JJ Thomson Avenue
Cambridge CB3 0HE, UK
E-mail:
Professor William T. Rhodes
Florida Atlantic University
Imaging Technology Center
Department of Electrical Engineering
777 Glades Road SE, Room 456
Boca Raton, FL 33431, USA
E-mail:
Professor H. Eugene Stanley
Boston University

Center for Polymer Studies
Department of Physics
590 Commonwealth Avenue, Room 204B
Boston, MA 02215, USA
E-mail:

www.pdfgrip.com


Rainer Dick

Advanced
Quantum
Mechanics
Materials and Photons
With 62 Figures

123
www.pdfgrip.com


Rainer Dick
University of Saskatchewan
Saskatoon, Saskatchewan
S7N5E2, Canada


ISSN 1868-4513
e-ISSN 1868-4521
ISBN 978-1-4419-8076-2

e-ISBN 978-1-4419-8077-9
DOI 10.1007/978-1-4419-8077-9
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011943751
c Springer Science+Business Media, LLC 2012
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed is
forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

www.pdfgrip.com


Preface
Quantum mechanics was invented in an era of intense and seminal scientific research between 1900 and 1928 (and in many regards continues to be developed
and expanded) because neither the properties of atoms and electrons, nor the
spectrum of radiation from heat sources could be explained by the classical
theories of mechanics, electrodynamics and thermodynamics. It was a major
intellectual achievement and a breakthrough of curiosity driven fundamental
research which formed quantum theory into one of the pillars of our present
understanding of the fundamental laws of nature. The properties and behavior of every elementary particle is governed by the laws of quantum theory.
However, the rule of quantum mechanics is not limited to atomic and subatomic scales, but also affects macroscopic systems in a direct and profound
manner. The electric and thermal conductivity properties of materials are determined by quantum effects, and the electromagnetic spectrum emitted by a
star is primarily determined by the quantum properties of photons. It is therefore not surprising that quantum mechanics permeates all areas of research

in advanced modern physics and materials science, and training in quantum
mechanics plays a prominent role in the curriculum of every major physics or
chemistry department.
The ubiquity of quantum effects in materials implies that quantum mechanics
also evolved into a major tool for advanced technological research. The construction of the first nuclear reactor in Chicago in 1942 and the development of
nuclear technology could not have happened without a proper understanding of
the quantum properties of particles and nuclei. However, the real breakthrough
for a wide recognition of the relevance of quantum effects in technology occured
with the invention of the transistor in 1948 and the ensuing rapid development
of semiconductor electronics. This proved once and for all the importance of
quantum mechanics for the applied sciences and engineering, only 22 years
after publication of the Schrăodinger equation! Electronic devices like transistors rely heavily on the quantum mechanical emergence of energy bands in
materials, which can be considered as a consequence of combination of many
atomic orbitals or as a consequence of delocalized electron states probing a
lattice structure. Today the rapid developments of spintronics, photonics and
nanotechnology provide continuing testimony to the technological relevance of
quantum mechanics.
As a consequence, every physicist, chemist and electrical engineer nowadays
has to learn aspects of quantum mechanics, and we are witnessing a time
v

www.pdfgrip.com


vi

Preface

when also mechanical and aerospace engineers are advised to take at least a
2nd year course, due to the importance of quantum mechanics for elasticity and

stability properties of materials. Furthermore, quantum information appears to
become inceasingly relevant for computer science and information technology,
and a whole new area of quantum technology will likely follow in the wake
of this development. Therefore it seems safe to posit that within the next
two generations, 2nd and 3rd year quantum mechanics courses will become as
abundant and important in the curricula of science and engineering colleges as
first and second year calculus courses.
Quantum mechanics continues to play a dominant role in particle physics and
atomic physics - after all, the Standard Model of particle physics is a quantum
theory, and the spectra and stability of atoms cannot be explained without
quantum mechanics. However, most scientists and engineers use quantum mechanics in advanced materials research. Furthermore, the dominant interaction
mechanisms in materials (beyond the nuclear level) are electromagnetic, and
many experimental techniques in materials science are based on photon probes.
The introduction to quantum mechanics in the present book takes this into
account by including aspects of condensed matter theory and the theory of
photons at earlier stages and to a larger extent than other quantum mechanics
texts. Quantum properties of materials provide neat and very interesting illustrations of time-independent and time-dependent perturbation theory, and
many students are better motivated to master the concepts of quantum mechanics when they are aware of the direct relevance for modern technology.
A focus on the quantum mechanics of photons and materials is also perfectly
suited to prepare students for future developments in quantum information
technology, where entanglement of photons or spins, decoherence, and time
evolution operators will be key concepts.
Other novel features of the discussion of quantum mechanics in this book
concern attention to relevant mathematical aspects which otherwise can only
be found in journal articles or mathematical monographs. Special appendices
include a mathematically rigorous discussion of the completeness of SturmLiouville eigenfunctions in one spatial dimension, an evaluation of the BakerCampbell-Hausdorff formula to higher orders, and a discussion of logarithms of
matrices. Quantum mechanics has an extremely rich and beautiful mathematical structure. The growing prominence of quantum mechanics in the applied
sciences and engineering has already reinvigorated increased research efforts
on its mathematical aspects. Both students who study quantum mechanics
for the sake of its numerous applications, as well as mathematically inclined

students with a primary interest in the formal structure of the theory should
therefore find this book interesting.
This book emerged from a quantum mechanics course which I had introduced
at the University of Saskatchewan in 2001. It should be suitable both for
advanced undergraduate and introductory graduate courses on the subject.
To make advanced quantum mechanics accessible to wider audiences which
might not have been exposed to standard second and third year courses on

www.pdfgrip.com


Preface

vii

atomic physics, analytical mechanics, and electrodynamics, important aspects
of these topics are briefly, but concisely introduced in special chapters and
appendices. The success and relevance of quantum mechanics has reached far
beyond the realms of physics research, and physicists have a duty to disseminate the knowledge of quantum mechanics as widely as possible.
Saskatoon, Saskatchewan, Canada

www.pdfgrip.com

Rainer Dick


www.pdfgrip.com


To the Students

Congratulations! You have reached a stage in your studies where the topics
of your inquiry become ever more interesting and more relevant for modern
research in basic science and technology.
Together with your professors, I will have the privilege to accompany you along
the exciting road of your own discovery of the bizarre and beautiful world of
quantum mechanics. I will aspire to share my own excitement that I continue
to feel for the subject and for science in general.
You will be introduced to many analytical and technical skills that are used
in everyday applications of quantum mechanics. These skills are essential in
virtually every aspect of modern research. A proper understanding of a materials science measurement at a synchrotron requires a proper understanding of
photons and quantum mechanical scattering, just like manipulation of qubits
in quantum information research requires a proper understanding of spin and
photons and entangled quantum states. Quantum mechanics is ubiquitous in
modern research. It governs the formation of microfractures in materials, the
conversion of light into chemical energy in chlorophyll or into electric impulses
in our eyes, and the creation of particles at the Large Hadron Collider.
Technical mastery of the subject is of utmost importance for understanding
quantum mechanics. Trying to decipher or apply quantum mechanics without
knowing how it really works in the calculation of wave functions, energy levels,
and cross sections is just idle talk, and always prone for misconceptions. Therefore we will go through a great many technicalities and calculations, because
you and I (and your professor!) have a common goal: You should become an
expert in quantum mechanics.
However, there is also another message in this book. The apparently exotic
world of quantum mechanics is our world. Our bodies and all the world around
us is built on quantum effects and ruled by quantum mechanics. It is not
apparent and only visible to the cognoscenti. Therefore we have developed a
mode of thought and explanation of the world that is based on classical pictures
– mostly waves and particles in mechanical interaction. This mode of thought
was sufficient for survivial of our species so far, and it culminated in a powerful
tool called classical physics. However, by 1900 those who were paying attention

had caught enough glimpses of the underlying non-classical world to embark
on the exciting journey of discovering quantum mechanics. Indeed, every single
atom in your body is ruled by the laws of quantum mechanics, and could not
even exist as a classical particle. The electrons that provide the light for your
ix

www.pdfgrip.com


x

To the Students

long nights of studying generate this light in stochastic quantum leaps from a
state of a single electron to a state of an electron and a photon. And maybe
the most striking example of all: There is absolutely nothing classical in the
sunlight that provides the energy for all life on Earth.
Quantum theory is not a young theory any more. The scientific foundations
of the subject were developed over half a century between 1900 and 1949,
and many of the mathematical foundations were even developed in the 19th
century. The steepest ascent in the development of quantum theory appeared
between 1924 and 1928, when matrix mechanics, Schrăodingers equation, the
Dirac equation and eld quantization were invented. I have included numerous
references to original papers from this period, not to ask you to read all those
papers – after all, the primary purpose of a textbook is to put major achievements into context, provide an introductory overview at an appropriate level,
and replace often indirect and circuitous original derivations with simpler explanations – but to honour the people who brought the then nascent theory
to maturity. Quantum theory is an extremely well established and developed
theory now, which has proven itself on numerous occasions. However, we still
continue to improve our collective understanding of the theory and its wide
ranging applications, and we test its predicitions and its probabilistic interpretation with ever increasing accuracy. The implications and applications of

quantum mechanics are limitless, and we are witnessing a time when many
technologies have reached their “quantum limit”, which is a misnomer for the
fact that any methods of classical physics are just useless in trying to describe
or predict the behavior of atomic scale devices. It is a “limit” for those who do
not want to learn quantum physics. For you, it holds the promise of excitement
and opportunity if you are prepared to work hard and if you can understand
the calculations.
Quantum mechanics combines power and beauty in a way that even supersedes advanced analytical mechanics and electrodynamics. Quantum mechanics is universal and therefore incredibly versatile, and if you have a sense for
mathematical beauty: the structure of quantum mechanics is breathtaking,
indeed.
I sincerely hope that reading this book will be an enjoyable and exciting experience for you.

To the Instructor
Dear Colleague,
as professors of quantum mechanics courses, we enjoy the privilege of teaching one of the most exciting subjects in the world. However, we often have
to do this with fewer lecture hours than were available for the subject in the
past, when at the same time we should include more material to prepare students for research or modern applications of quantum mechanics. Furthermore,
students have become more mobile between universities (which is good) and
between academic programs (which can have positive and negative implica-

www.pdfgrip.com


To the Students

xi

tions). Therefore we are facing the task to teach an advanced subject to an
increasingly heterogeneous student body with very different levels of preparation. Nowadays the audience in a fourth year undergraduate or beginning
graduate course often includes students who have not gone through a course

on Lagrangian mechanics, or have not seen the covariant formulation of electrodynamics in their electromagnetism courses. I deal with this problem by
including one special lecture on each topic in my quantum mechanics course,
and this is what Appendices A and B are for. I have also tried to be as inclusive
as possible without sacrificing content or level of understanding by starting at
a level that would correspond to an advanced second year Modern Physics
or Quantum Chemistry course and then follow a steeply ascending route that
takes the students all the way from Planck’s law to the photon scattering
tensor.
The selection and arrangement of topics in this book is determined by the
desire to develop an advanced undergraduate and introductory gaduate level
course that is useful to as many students as possible, in the sense of giving
them a head start into major current research areas or modern applications of
quantum mechanics without neglecting the necessary foundational training.
There is a core of knowledge that every student is expected to know by heart
after having taken a course in quantum mechanics. Students must know the
Schrăodinger equation. They must know how to solve the harmonic oscillator
and the Coulomb problem, and they must know how to extract information
from the wave function. They should also be able to apply basic perturbation
theory, and they should understand that a wave function x|ψ(t) is only one
particular representation of a quantum state |ψ(t) .
In a North American physics program, students would traditionally learn all
these subjects in a 300-level Quantum Mechanics course. Here these subjects
are discussed in Chapters 1-7 and 9. This allows the instructor to use this book
also in 300-level courses or introduce those chapters in a 400-level or graduate
course if needed. Depending on their specialization, there will be an increasing
number of students from many different science and engineering programs
who will have to learn these subjects at M.Sc. or beginning Ph.D. level before
they can learn about photon scattering or quantum effects in materials, and
catering to these students will also become an increasingly important part of
the mandate of physics departments. Including chapters 1-7 and 9 with the

book is part of the philosophy of being as inclusive as possible to disseminate
knowledge in advanced quantum mechanics as widely as possible.
Additional training in quantum mechanics in the past traditionally focused
on atomic and nuclear physics applications, and these are still very important
topics in fundamental and applied science. However, a vast number of our
current students in quantum mechanics will apply the subject in materials
science in a broad sense encompassing condensed matter physics, chemistry
and engineering. For these students it is beneficial to see Bloch’s theorem,
Wannier states, and basics of the theory of covalent bonding embedded with
their quantum mechanics course. Another important topic for these students

www.pdfgrip.com


xii

To the Students

is quantization of the Schrăodinger eld. Indeed, it is also useful for students
in nuclear and particle physics to learn quantization of the Schrăodinger eld
because it makes quantization of gauge fields and relativistic matter fields so
much easier if they know quantum field theory in the non-relativistic setting.
Furthermore, many of our current students will use or manipulate photon
probes in their future graduate and professional work. A proper discussion
of photon-matter interactions is therefore also important for a modern quantum mechanics course. This should include minimal coupling, quantization of
the Maxwell field, and applications of time-dependent perturbation theory for
photon absorption, emission and scattering.
Students should also know the Klein-Gordon and Dirac equations after completion of their course, not only to understand that Schrăodingers equation is
not the final answer in terms of wave equations for matter particles, but to
understand the nature of relativistic corrections like Pauli or Rashba terms.

The scattering matrix is introduced as early as possible in terms of matrix
elements of the time evolution operator on states in the interaction picture,
Sf i (t, t ) = f |UD (t, t )|i , cf. equation (13.20). This representation of the scattering matrix appears so naturally in ordinary time-dependent perturbation
theory that it makes no sense to defer the notion of an S-matrix to the discussion of scattering in quantum field theory with two or more particles in the
initial state. It actually mystifies the scattering matrix to defer its discussion
until field quantization has been introduced. On the other hand, introducing
the scattering matrix even earlier in the framework of scattering off static
potentials is counterproductive, because its natural and useful definition as
matrix elements of a time evolution operator cannot properly be introduced
at that level, and the notion of the scattering matrix does not really help with
the calculation of cross sections for scattering off static potentials.
I have also emphasized the discussion of the various roles of transition matrix
elements depending on whether the initial or final states are discrete or continuous. It helps students to understand transition probabilities, decay rates,
absorption cross sections and scattering cross sections if the discussion of these
concepts is integrated in one chapter, cf. Chapter 13. Furthermore, I have put
an emphasis on canonical field quantization. Path integrals provide a very elegant description for free-free scattering, but bound states and energy levels,
and basic many-particle quantum phenomena like exchange holes are very efficiently described in the canonical formalism. Feynman rules also appear more
intuitive in the canonical formalism of explicit particle creation and annihilation.
The core advanced topics in quantum mechanics that an instructor might want
to cover in a traditional 400-level or introductory graduate course are included
with Chapters 8, 11-13, 15-18, and 21. However, instructors of a more inclusive
course for general science and engineering students should include materials
from Chapters 1-7 and 9, as appropriate.

www.pdfgrip.com


To the Students

xiii

The direct integration of training in quantum mechanics with the foundations
of condensed matter physics, field quantization, and quantum optics is very
important for the advancement of science and technology. I hope that this
book will help to achieve that goal. I would greatly appreciate your comments
and criticism. Please send them to

www.pdfgrip.com


www.pdfgrip.com


Contents
1 The Need for Quantum Mechanics
1.1 Electromagnetic spectra and evidence
for discrete energy levels . . . . . . . . . . .
1.2 Blackbody radiation and Planck’s law . . . .
1.3 Blackbody spectra and photon fluxes . . . .
1.4 The photoelectric effect . . . . . . . . . . . .
1.5 Wave-particle duality . . . . . . . . . . . . .
1.6 Why Schrăodingers equation? . . . . . . . .
1.7 Interpretation of Schrăodingers wave function
1.8 Problems . . . . . . . . . . . . . . . . . . . .

1
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.

.
.
.
.

1
3
8
14
15
16
18
22

2 Self-adjoint Operators and Eigenfunction Expansions
25
2.1 The δ function and Fourier transforms . . . . . . . . . . . . . . 25
2.2 Self-adjoint operators and completeness of eigenstates . . . . . . 30
2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Simple Model Systems
3.1 Barriers in quantum mechanics . . . . . . . . . .
3.2 Quantum wells, quantum wires and quantum dots
3.3 The attractive δ function potential . . . . . . . .
3.4 Evolution of free Schrăodinger wave packets . . . .
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.

.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

35
35
42
45

47
52

4 Notions from Linear Algebra and Bra-Ket Notation
4.1 Notions from linear algebra . . . . . . . . . . . . . . . .
4.2 Bra-ket notation in quantum mechanics . . . . . . . . . .
4.3 The adjoint Schrăodinger equation and the virial theorem
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

57

58
66
70
72

. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
coordinate systems

.
.
.
.

75
75
79
81
82

5 Formal Developments
5.1 Uncertainty relations . . . . . . . . . . . .
5.2 Energy representation . . . . . . . . . . .
5.3 Dimensions of states . . . . . . . . . . . .
5.4 Gradients and Laplace operators in general

.
.
.

.
.

.
.
.
.
.

.
.
.
.
.

xv

www.pdfgrip.com


xvi

Contents
5.5
5.6

Separation of differential equations . . . . . . . . . . . . . . . . 86
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Harmonic Oscillators and Coherent States

6.1 Basic aspects of harmonic oscillators . . . . . . . . . . . .
6.2 Solution of the harmonic oscillator by the operator method
6.3 Construction of the states in the x-representation . . . . .
6.4 Lemmata for exponentials of operators . . . . . . . . . . .
6.5 Coherent states . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

91
91

92
94
96
98
104

7 Central Forces in Quantum Mechanics
107
7.1 Separation of center of mass motion and relative motion . . . . 107
7.2 The concept of symmetry groups . . . . . . . . . . . . . . . . . 110
7.3 Operators for kinetic energy and angular momentum . . . . . . 111
7.4 Defining representation of the three-dimensional rotation group 113
7.5 Matrix representations of the rotation group . . . . . . . . . . . 114
7.6 Construction of the spherical harmonic functions . . . . . . . . . 116
7.7 Basic features of motion in central potentials . . . . . . . . . . . 120
7.8 Free spherical waves: the free particle with sharp Mz , M 2 . . . . 121
7.9 Bound energy eigenstates of the hydrogen atom . . . . . . . . . 124
7.10 Spherical Coulomb waves . . . . . . . . . . . . . . . . . . . . . . 131
7.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8 Spin and Addition of Angular Momentum Type Operators
8.1 Spin and magnetic dipole interactions . . . . . . . . . . . . . .
8.2 Transformation of wave functions under rotations . . . . . . .
8.3 Addition of angular momentum like quantities . . . . . . . . .
8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.

139
139
142
144
150

9 Stationary Perturbations in Quantum Mechanics
151
9.1 Time-independent perturbation theory without degeneracies . . 151
9.2 Time-independent perturbation theory with degeneracies . . . . 156
9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10 Quantum Aspects of Materials I
10.1 Bloch’s theorem . . . . . . . . . . . . . . .
10.2 Wannier states . . . . . . . . . . . . . . .
10.3 Time-dependent Wannier states . . . . . .
10.4 The Kronig-Penney model . . . . . . . . .
10.5 kp perturbation theory and effective mass
10.6 Problems . . . . . . . . . . . . . . . . . . .

www.pdfgrip.com

.
.
.
.
.
.

.
.

.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.

.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

163
163
166
169
170
174
176


Contents

xvii

11 Scattering Off Potentials
11.1 The free energy dependent Green’s function . .

11.2 Potential scattering in the Born approximation .
11.3 Scattering off a hard sphere . . . . . . . . . . .
11.4 Rutherford scattering . . . . . . . . . . . . . . .
11.5 Problems . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.

.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

183

185
187
190
194
198

12 The
12.1
12.2
12.3
12.4
12.5

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

199
200
202
204
205
207

13 Time-dependent Perturbations in Quantum Mechanics
13.1 Pictures of quantum dynamics . . . . . . . . . . . . . . .
13.2 The Dirac picture . . . . . . . . . . . . . . . . . . . . . .
13.3 Transitions between discrete states . . . . . . . . . . . .
13.4 Transitions from discrete states into continuous states . .
13.5 Transitions from continuous states into discrete states . .
13.6 Transitions between continuous states – scattering . . . .
13.7 Expansion of the scattering matrix to higher orders . . .
13.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

209
209
215
218
223
230
232
237
238

Density of States
Counting of oscillation modes . . .
The continuum limit . . . . . . . .
Density of states per unit of energy
Density of states in radiation . . . .
Problems . . . . . . . . . . . . . . .

.
.
.
.
.

.
.
.
.
.

.

.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

14 Path Integrals in Quantum Mechanics

241
14.1 Time evolution in the path integral formulation . . . . . . . . . 242
14.2 Path integrals in scattering theory . . . . . . . . . . . . . . . . . 247
14.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
15 Coupling to Electromagnetic Fields
15.1 Electromagnetic couplings . . . . .
15.2 Stark effect and static polarizability
15.3 Dynamical polarizability tensors . .
15.4 Problems . . . . . . . . . . . . . . .

. . . . .
tensors
. . . . .
. . . . .

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

255
255
261
263
270

16 Principles of Lagrangian Field Theory
273
16.1 Lagrangian field theory . . . . . . . . . . . . . . . . . . . . . . . 273
16.2 Symmetries and conservation laws . . . . . . . . . . . . . . . . . 276
16.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
17 Non-relativistic Quantum Field Theory
17.1 Quantization of the Schrăodinger eld . . . . . . . . . . . .
17.2 Time evolution for time-dependent Hamiltonians . . . . . .
17.3 The connection between first and second quantized theory
17.4 The Dirac picture in quantum field theory . . . . . . . . .
17.5 Inclusion of spin . . . . . . . . . . . . . . . . . . . . . . . .

www.pdfgrip.com

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

283
284
291
292
296
299


xviii
17.6
17.7
17.8

17.9

Contents
Two-particle interaction potentials and equations of
Expectation values and exchange terms . . . . . . .
From many particle theory to second quantization .
Problems . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.

.
.
.
.

.
.
.
.

303
307
309
311

18 Quantization of the Maxwell Field: Photons
18.1 Lagrange density and mode expansion for the Maxwell field .

18.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Coherent states of the electromagnetic field . . . . . . . . . .
18.4 Photon coupling to relative motion . . . . . . . . . . . . . .
18.5 Photon emission rates . . . . . . . . . . . . . . . . . . . . .
18.6 Photon absorption . . . . . . . . . . . . . . . . . . . . . . .
18.7 Stimulated emission of photons . . . . . . . . . . . . . . . .
18.8 Photon scattering . . . . . . . . . . . . . . . . . . . . . . . .
18.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

321

321
327
329
330
331
339
342
344
352

19 Quantum Aspects of Materials II
19.1 The Born-Oppenheimer approximation
19.2 Covalent bonding . . . . . . . . . . . .
19.3 Bloch and Wannier operators . . . . .
19.4 The Hubbard model . . . . . . . . . .
19.5 Vibrations in molecules and lattices . .
19.6 Quantized lattice vibrations – phonons
19.7 Electron-phonon interactions . . . . . .
19.8 Problems . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.

.

.
.
.
.
.
.
.

355
356
359
368
372
374
385
389
393

.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

motion
. . . .
. . . .
. . . .

.
.
.
.

.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.

.
.
.
.
.
.
.
.

20 Dimensional Effects in Low-dimensional Systems
397
20.1 Quantum mechanics in d dimensions . . . . . . . . . . . . . . . 397
20.2 Inter-dimensional effects in interfaces and thin layers . . . . . . 403
20.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
21 Klein-Gordon and Dirac Fields
21.1 The Klein-Gordon equation . . . . . . . . . . . . . . .
21.2 Klein’s paradox . . . . . . . . . . . . . . . . . . . . . .
21.3 The Dirac equation . . . . . . . . . . . . . . . . . . . .
21.4 Energy-momentum tensor for quantum electrodynamics
21.5 The non-relativistic limit of the Dirac equation . . . . .
21.6 Two-particle scattering cross sections . . . . . . . . . .
21.7 Photon scattering by free electrons . . . . . . . . . . .
21.8 Møller scattering . . . . . . . . . . . . . . . . . . . . .
21.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

413
413
419
423
429
433
435
440
450
458

Appendix A: Lagrangian Mechanics

461

Appendix B: The Covariant Formulation of Electrodynamics

467

www.pdfgrip.com


Contents

xix

Appendix C: Completeness of Sturm-Liouville Eigenfunctions

483

Appendix D: Properties of Hermite Polynomials

497

Appendix E: The Baker-Campbell-Hausdorff Formula

499

Appendix F: The Logarithm of a Matrix

503

Appendix G: Dirac

507

matrices

Appendix H: Spinor representations of the Lorentz group

517

Appendix I: Green’s functions in d dimensions

525

Bibliography

543

Index

547

www.pdfgrip.com


www.pdfgrip.com


Chapter 1
The Need for Quantum
Mechanics

1.1

Electromagnetic spectra and evidence
for discrete energy levels

Quantum mechanics was initially invented because classical mechanics,
thermodynamics and electrodynamics provided no means to explain the properties of atoms, electrons, and electromagnetic radiation. Furthermore, it
became clear after the introduction of Schrăodingers equation and the quantization of Maxwells equations that we cannot explain any physical property
of matter and radiation without the use of quantum theory. We will see a lot
of evidence for this in the following chapters. However, in the present chapter
we will briefly and selectively review the early experimental observations and
developments which led to the development of quantum mechanics over a
period of intense research between 1900 and 1928.
The first evidence that classical physics was incomplete appeared in unexpected properties of electromagnetic spectra. Thin gases of atoms or molecules
emit line spectra which contradict the fact that a classical system of electric
charges can oscillate at any frequency, and therefore can emit radiation of any
frequency. This was a major scientific puzzle from the 1850s until the inception
of the Schrăodinger equation in 1926.
Contrary to a thin gas, a hot body does emit a continuous spectrum, but even
those spectra were still puzzling because the shape of heat radiation spectra
could not be explained by classical thermodynamics and electrodynamics. In
fact, classical physics provided no means at all to predict any sensible shape
for the spectrum of a heat source! But at last, hot bodies do emit a continuous
spectrum and therefore, from a classical point of view, their spectra are not
quite as strange and unexpected as line spectra. It is therefore not surprising
that the first real clues for a solution to the puzzles of electromagnetic spectra
emerged when Max Planck figured out a way to calculate the spectra of heat
sources under the simple, but classically extremely counterintuitive assumption
R. Dick, Advanced Quantum Mechanics: Materials and Photons,
Graduate Texts in Physics, DOI 10.1007/978-1-4419-8077-9 1,

c Springer Science+Business Media, LLC 2012

www.pdfgrip.com

1


2

Chapter 1. The Need for Quantum Mechanics

that the energy in heat radiation of frequency f is quantized in integer multiples
of a minimal energy quantum hf ,
E = nhf,

n ∈ N.

(1.1)

The constant h that Planck had introduced to formulate this equation became
known as Planck’s constant and it could be measured from the shape of heat radiation spectra. A modern value is h = 6.626×10−34 J · s = 4.136×10−15 eV · s.
We will review the puzzle of heat radiation and Planck’s solution in the next
section, because Planck’s calculation is instructive and important for the understanding of incandescent light sources and it illustrates in a simple way how
quantization of energy levels yields results which are radically different from
predictions of classical physics.
Albert Einstein then pointed out that equation (1.1) also explains the photoelectric effect. He also proposed that Planck’s quantization condition is not a
property of any particular mechanism for generation of electromagnetic waves,
but an intrinsic property of electromagnetic waves. However, once equation
(1.1) is accepted as an intrinsic property of electromagnetic waves, it is a small
step to make the connection with line spectra of atoms and molecules and conclude that these line spectra imply existence of discrete energy levels in atoms

and molecules. Somehow atoms and molecules seem to be able to emit radiation only by jumping from one discrete energy state into a lower discrete energy
state. This line of reasoning, combined with classical dynamics between electrons and nuclei in atoms then naturally leads to the Bohr-Sommerfeld theory
of atomic structure. This became known as old quantum theory.
Apparently, the property which underlies both the heat radiation puzzle and
the puzzle of line spectra is discreteness of energy levels in atoms, molecules,
and electromagnetic radiation. Therefore, one major motivation for the development of quantum mechanics was to explain discrete energy levels in atoms,
molecules, and electromagnetic radiation.
It was Schrăodingers merit to find an explanation for the discreteness of energy
levels in atoms and molecules through his wave equation1 ( ≡ h/2π)
2
∂
ψ(x, t) = −
Δψ(x, t) + V (x)ψ(x, t).
(1.2)
∂t
2m
A large part of this book will be dedicated to the discussion of Schrăodingers
equation. An intuitive motivation for this equation will be given in Section 1.6.
Ironically, the fundamental energy quantization condition (1.1) for electromagnetic waves, which precedes the realization of discrete energy levels in atoms
and molecules, cannot be derived by solving a wave equation, but emerges from
the quantization of Maxwell’s equations. This is at the heart of understanding
photons and the quantum theory of electromagnetic waves. We will revisit this
issue in Chapter 18. However, we can and will discuss already now the early
quantum theory of the photon and what it means for the interpretation of
spectra from incandescent sources.

i

1

E. Schrăodinger, Annalen Phys. 386, 109 (1926).

www.pdfgrip.com


1.2. Blackbody radiation and Planck’s law

1.2

3

Blackbody radiation and Planck’s law

Historically, Planck’s deciphering of the spectra of incandescent heat and light
sources played a key role for the development of quantum mechanics, because
it included the first proposal of energy quanta, and it implied that line spectra
are a manifestation of energy quantization in atoms and molecules. Planck’s
radiation law is also extremely important in astrophysics and in the technology
of heat and light sources.
Generically, the heat radiation from an incandescent source is contaminated
with radiation reflected from the source. Pure heat radiation can therefore only
be observed from a non-reflecting, i.e. perfectly black body. Hence the name
blackbody radiation for pure heat radiation. Physicists in the late 19th century
recognized that the best experimental realization of a black body is a hole in
a cavity wall. If the cavity is kept at temperature T , the hole will emit perfect
heat radiation without contamination from any reflected radiation.
Suppose we have a heat radiation source (or thermal emitter) at temperature
T . The power per area radiated from a thermal emitter at temperature T
is denoted as its exitance (or emittance) e(T ). In the blackbody experiments
e(T ) · A is the energy per time leaking through a hole of area A in a cavity

wall.
To calculate e(T ) as a function of the temperature T , as a first step we need
to find out how it is related to the density u(T ) of energy stored in the heat
radiation. One half of the radiation will have a velocity component towards the
hole, because all the radiation which moves under an angle ϑ ≤ π/2 relative to
the axis going through the hole will have a velocity component v(ϑ) = c cos ϑ
in the direction of the hole. To find out the average speed v of the radiation
in the direction of the hole, we have to average c cos ϑ over the solid angle
Ω = 2π sr of the forward direction 0 ≤ ϕ ≤ 2π, 0 ≤ ϑ ≤ π/2:
v=

c
2π

2π

π/2

dϕ
0

0

c
dϑ sin ϑ cos ϑ = .
2

The effective energy current density towards the hole is energy density moving
in forward direction × average speed in forward direction:
u(T ) c

c
= u(T ) ,
2 2
4
and during the time t an amount of energy
c
E = u(T ) tA
4
will escape through the hole. Therefore the emitted power per area E/(tA) =
e(T ) is
c
e(T ) = u(T ) .
4

(1.3)

www.pdfgrip.com


4

Chapter 1. The Need for Quantum Mechanics

However, Planck’s radiation law is concerned with the spectral exitance e(f, T ),
which is defined in such a way that
f2

e[f1 ,f2 ] (T ) =

df e(f, T )

f1

is the power per area emitted in radiation with frequencies f1 ≤ f ≤ f2 .
In particular, the total exitance is
∞

e(T ) = e[0,∞] (T ) =

df e(f, T ).
0

Operationally, the spectral exitance is the power per area emitted with
frequencies f ≤ f ≤ f + Δf , and normalized by the width Δf of the
frequency interval,
e(f, T ) = lim

Δf →0

e[f,f +Δf ] (T )
e[0,f +Δf ] − e[0,f ] (T )
∂
= lim
=
e[0,f ] (T ).
Δf →0
Δf
Δf
∂f

The spectral exitance e(f, T ) can also be denoted as the emitted power per area

and per unit of frequency or as the spectral exitance in the frequency scale.
The spectral energy density u(f, T ) is defined in the same way. If we measure
the energy density u[f,f +Δf ] (T ) in radiation with frequency between f and
f + Δf , then the energy per volume and per unit of frequency (i.e. the spectral
energy density in the frequency scale) is
u[f,f +Δf ] (T )
∂
=
u[0,f ] (T ),
Δf →0
Δf
∂f

u(f, T ) = lim

(1.4)

and the total energy density in radiation is
∞

u(T ) =

df u(f, T ).
0

The equation e(T ) = u(T )c/4 also applies separately in each frequency interval [f, f + Δf ], and therefore must also hold for the corresponding spectral
densities,
c
e(f, T ) = u(f, T ) .
4

(1.5)

The following facts were known before Planck’s work in 1900.
• The prediction from classical thermodynamics for the spectral exitance
e(f, T ) (Rayleigh-Jeans law) was wrong, and actually non-sensible!
• The exitance e(T ) satisfies Stefan’s law (Stefan, 1879; Boltzmann, 1884)
e(T ) = σT 4 ,
with the Stefan-Boltzmann constant
σ = 5.6704 × 10−8

W
.
m2 K4

www.pdfgrip.com